In algebra, there is a certain order in which operations must be done. If you’re reading this site, there’s a fair chance that you’re so familiar with this concept that it’s basically subconscious. If not, you’ve probably heard of the mnemonic BEDMAS, meaning brackets, exponents, division/multiplication, and addition/subtraction. These rules are formulated so that there is logical consistency in algebra.
If I wanted you to calculate a bunch of derivatives for me, I could simply show you the algorithmic approach to the process and leave you to do it. With some practice, you’d get better and better at taking derivatives and could even become better than a mathematician.